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A336614 Number of n X n (0,1)-matrices A over the reals such that A^2 is the transpose of A. 3
1, 2, 4, 10, 32, 112, 424, 1808, 8320, 40384, 210944, 1170688, 6783616, 41411840, 265451008, 1765520128, 12227526656, 88163295232, 656548065280, 5054719287296, 40261285543936, 330010835894272, 2783003772452864, 24166721466204160, 215318925894909952, 1966855934183800832 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Peter Luschny, Jun 04 2021: (Start)

a(n) = n! * [x^n] exp(x*(x^2 + 6)/3).

a(n) = 2*a(n - 1) + (n^2 - 3*n + 2)*a(n - 3) for n >= 3.

a(n) = Sum_{k=0..n/3} (2^(n-3*k)*n!)/(3^k*k!*(n-3*k)!).

a(n) = 2^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [], -9/8).

[The above formulas, first stated as conjectures, were proved by mjqxxxx at Mathematics Stack Exchange, see link.] (End)

LINKS

Table of n, a(n) for n=0..25.

mjqxxxx, Proof of conjectured formulas, Mathematics Stack Exchange.

FORMULA

a(n) = A336174(n) + A000079(n).

EXAMPLE

a(3) = A336174(3) + A000079(3) = 2 + 8 = 10.

MAPLE

a := n -> add((2^(n - 3*k)*n!)/(3^k*k!*(n - 3*k)!), k=0..n/3):

seq(a(n), n=0..25); # Peter Luschny, Jun 05 2021

PROG

(PARI) m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)

a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~)

for(n = 0, 9, print1(a(n), ", "))

(Python)

from itertools import product

from sympy import Matrix

def A336614(n):

    c = 0

    for d in product((0, 1), repeat=n*n):

        M = Matrix(d).reshape(n, n)

        if M*M == M.T:

            c += 1

    return c # Chai Wah Wu, Sep 29 2020

CROSSREFS

Row sums of A344912.

Cf. A000079, A001471, A336174.

Sequence in context: A296003 A263662 A151400 * A071954 A120017 A000736

Adjacent sequences:  A336611 A336612 A336613 * A336615 A336616 A336617

KEYWORD

nonn

AUTHOR

Torlach Rush, Jul 27 2020

EXTENSIONS

More terms from Peter Luschny, Jun 05 2021

STATUS

approved

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Last modified September 22 22:13 EDT 2021. Contains 347608 sequences. (Running on oeis4.)